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I'm looking for a formula or quick trick for computing the branch locus of a hypersurface under projection. My specific case:

Let $V\subset\mathbb{P}^3$ be a surface of degree $d$, $p\in\mathbb{P}^3$ (it's actually a node of $V$, if that's relevant), and I'm attempting to compute the branch locus of the projection $V\dashrightarrow \mathbb{P}^2$, which should be a plane curve.

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Perhaps this could be helpful:…^2&source=bl&ots=PTrmd4B3uF&sig=4JADnIAEff_R2YoKAFeszP_Gbc0&hl‌​=en&ei=XstETbPgGMGB8ga4htSZAg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCMQ6AE‌​wAg#v=onepage&q=computing%20the%20branch%20locus%20V%20of%20P%5E2&f=false – PEV Jan 30 '11 at 2:25

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